The vector measures whose range is strictly convex

Let mu be a measure on a measure space (X, Lambda) with values in R-n and f be the density of mu with respect to its total variation. We show that the range R(mu)= {mu(E) : E is an element of Lambda} of mu is strictly convex if and only if the determinant det[f(x(1)),..,f(x(n))] is nonzero a.e, on X-n. We apply the result to a class of measeres containing those that are generated by Chebyshev systems. (C) 1999 Academic Press.